— Twitter Support (@Support) June 18, 2014
Animated GIFs will be shared with the same pic.twitter.com links the service uses for its native image uploads, and they will be animated inline.
Right now, clicking a pic.twitter.com GIF link in Tweetbot doesn't open the GIF but redirects to the same tweet. It's not clear at this point whether GIFs are supported in the Twitter API and if third-party developers will be able to display animated GIFs in their Twitter clients soon.
Update: Right after publishing, Twitter confirmed that GIFs will be supported in the API. Documentation is available here.
— TwitterDev (@TwitterDev) June 18, 2014